In this talk I will discuss the local and global conjectures for some strongly tempered spherical varieties. Both conjectures are very similar to the Gan-Gross-Prasad models. More specifically, globally the square of the period integrals should be related to the central value of some L-functions of symplectic type. Locally each tempered L-packet should contain a unique distinguished element with multiplicity one and the unique distinguished element should be determined by certain epsilon factors (i.e. epsilon dichotomy). I will also discuss the proof of the local conjecture in many cases. This is a joint work with Lei Zhang.